Re: Well Ordering the Reals

Ross A. Finlayson wrote:
> David McAnally wrote:
> > D.McAnally@i'm_a_gnu.uq.net.au (David McAnally) writes:
> >
> > >"Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> writes:
> >
> > >>David McAnally wrote:
> > >>> "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> writes:
> > >>>
> > >>> >There is no universe in ZF. (Shrug.)
> > >>>
> > >>> You keep claiming this, but you have never given a genuine proof of your
> > >>> claim.
> > >>>
> > >>> -----
> >
> > >>Hi Dave,
> >
> > >>Hey someone was asking for you over on sci.physics, they're trying to
> > >>figure out "infinity".
> >
> > >>Are you saying that there is a universe in ZF? There's not a universal
> > >>set, and objects in ZF are sets. Is that concise enough for you, as an
> > >>obvious logical progression, that no object in ZF is the universe?
> >
> > >I read your comment along the lines of your previously stated conviction
> > >that there is no model of ZF. You have stated many times that ZF is
> > >inconsistent, and you have never proffered a proof of this inconsistency.
> > >I misread your statement as a statement to that effect, because I was so
> > >used to seeing you make those claims.
> >
> > >There is no set in ZF which in the universe in a model (i.e. for any set
> > >in ZF, there is a set which is not an element of that set). The Axiom of
> > >Separation is sufficient to contradict the existence of such a set.
> >
> > This last paragraph should have read:
> >
> > There is no set in ZF which is the universe in a model (i.e. for any set
> > in ZF, there is a set which is not an element of that set). The Axiom of
> > Separation is sufficient to contradict the existence of such a universal
> > set.
> >
> > -----
>
> Hi David,
>
> Well, there's basicallly unrestricted comprehension, the domain
> principle. I think that implies the existence of a universal set, as
> did Cantor. Via empty, union and powerset, the sets are all
> hereditarily finite, but infinity posits the existence of basically a
> non-hereditarily finite set. Thus, with a non hereditarily finite set,
> via union and powerset, there are thus via that unrestriction of
> comprehension a set of all hereditarily finite sets that is not itself
> hereditarily finite. That's just to exhibit an unrestriction of
> comprehension.
>
> You agree there is no universe in ZF, that to me implies there is no
> predicate in ZF that evaluates to true, ie for any x, if true=true x E
> X, because in quantification over sets that seems to imply a universe,
> because of the fact that the only objects in the theory are sets.
> Basically infinity is at odds with regularity. The universal
> quantifier is quantifying over the universe.
>
> You mention separation, basically you refer to Russell's paradox, or
> the specification of the Russell set, that you could separate a
> universal set into those A that contain themselves and those B that
> don't, and if A E A, then B ~E B. Then, if B ~E B, then it doesn't, so
> it fits the predicate of B of those sets that don't, and B E B. Then B
> doesn't fit the predicate of not containing itself, yet B E B. It's
> the predicate of sets that contain themselves, so B E A, and B is in
> both sets, yet neither. It's the Russell paradox, and leads to
> separation anxiety.
>
> Consider sets that are hereditarily finite, and those that are not. If
> a set contains everything that's hereditarily finite, it's not
> hereditarily finite. The set containing that is hereditarily finite,
> it only has one element, yet variously as an ordinal N+1 and not. Is
> that wrong?
>
> That can lead to the notion of an anti-foundation where every pure set
> contains itself. That's seemingly indefensible, but I'm not the first
> person to posit that {} E {} or N E N, or even that U E O, for O Ord,
> zero, or both, although in general Ord. Then the set of all sets not
> containing themselves would be the empty set, which would unfortunately
> be containing itself, so there's the same problem, except the predicate
> is short-circuited. What if any infinite set contained itself, then
> the set of all sets not containing themselves would be the set of
> finite sets, which not being finite would contain itself, shuffling
> itself out of the Russell set, bijecting to a subset of itself, N E N+1
> and N is a subset of N+1.
>
> It's like the one-point compactification of the naturals, or
> hypernaturals and the transfer principle, N E N. Analysts are finding
> use of these notions like oo E R.
>
> So, yeah, having a universe as a set leads to the Russell paradox. Not
> having a universe in ZF leads to there being no universe, and the
> axioms discuss all sets. In von Neumann-Bernays-Goedel, there are the
> axiom schema, but then there would be no universe of those, either.
>
> Basically there's a difference between "for each" and "for all", but
> they're both the universal quantifier. A bucket of apples is not an
> apple, but it's not a pure set either.
>
> Where ZF doesn't have a universe, in cosmology and other applied
> domains there is definitely a universe, including all our concepts and
> discussions and talks about it.
>
> Oh, they were asking on sci.physics.relativity.
>
> Ross


Hi,

So, if infinite sets are irregular, and finite sets are not, then,
separating according to Russell's predicate the set of all sets that
contain themselves, the infinite (I initially wrote empty) sets, and
the sets of all sets that do not contain themselves, the finite sets,
has the set of all finite sets being an infinite set, which somehow
mathemagically contains itself.

The question then, is how does an infinite set lead to itself being not
only a subset, as is so for each set, but an element.

There are some notions of having infinite elements in the natural
numbers. For example, consider Zeno, half and then half again and then
half again.

Sum_i=1^n 1/2^n

The sum for i from 1 to n (in N) of 1 over 2 to the n.

For no finite natural integer n is that sum equal to one. As a binary
expansion, it's a finite string of ones after the radix. Now,
everybody here has heard that .111... = 1, but only for an infinite
number of digits in the expansion, not for any finite number, and they
are very much different representations.

lim n->oo Sum_i=1^n 1/2^n = 1

The limit as n goes to infinity of the sum for i from 1 to n (in N) of
1 over 2 to the n is equal to one.

The limit is used, basically via induction as n increases the sum
always gets closer to one, and eventually gets larger than the sum for
any value of n that is less than one, and is never greater than one.
While that is so, a more compact notation as this represents much the
same notion.

Sum_i=1^oo 1/2^n = 1

The sum for i from 1 to infinity of 1 over 2 to the n is equal to one.

For some, the point of having an actual solution instead of an
approximation, even in this case where the approximation is exact,
leads to such notions as the hypernaturals, a form of nonstandard
integers that basically said to infinitely many infinite integers after
all the finite integers. For others, there is the notion of a
projective extension of sorts or compactification of the domain of
natural numbers, the point at infinity, with basically having infinity,
one infinite element, and that's all, in the domain. Differences
between those can be seen as similar to the differences between
Newton's infinitesimals, with the fluxions for each fluent and each
fluxion being a fluent, and Leibniz' infinitesimals, the differential,
the sum of which as a constant from one to infinity is equal to one.

So, in this consideration of a set of all sets, or universe in a theory
where the only objects are sets thus any universal object would be a
set, one notion to consider is how in the higher levels of mathematics
assumptions about an infinite element, N in N, are intuitively used and
as not abused provide exact, correct, results.

Immanuel Kant, he discusses the Ding-an-Sich, the Thing-in-Itself, as
an intuitive notion. Some fringe physical theories have the universe
containing itself.

The notions of N E N, or +-oo E R, do find some use, already. Is that
use unrigorous? Perhaps another notion is that the sputniks of
quantification, of sorts, serve as means towards resolution of the
Russell paradox where the universal quantifier demands a universal set
in a set theory.

Then, there's still the notion of that object being its own powerset,
as is addressed in a variety of theories, theories with non-sets
objects, anti-foundational theories, pure set theories.

In to well order the reals, this is part of a separate discussion, or
separation discussion, in well-ordering the reals by Cantor's first
there are adjacent points in the reals, eg Leibniz' differential, and
the reals are thus not shown uncountable, ordering sensitive but not
uncountable, by Cantor's first, or the reals are not a set.

So, well-order the reals.

Ross

.

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